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I think they are pretty useful. Especially when you use them to filter out by checking (-).
For example, if you are looking for games that play with 7 people but you don't want to see party games. Or a game about animals, but not a children's game. Or games set in ancient Rome, but not wargames.
I never look at the Type and Category fields, my eyes are just not drawn there.
What I do look at all the time is the Mechanism field. That's one of the first things I check when finding out about a game for the first time. I also use that field extensively in searches. It's fun to do an advanced search for combinations of different mechanisms. For example what are some co-op trick-taking games besides Crew and Fox in the Forest Duet?
Yeah, mechanism is a lot more objective way to classify board games. We can argue all day about, say, what exactly a wargame is, but what games have hex grids, area control, and resource managements is a lot easier to agree on.
They are useful in combinations. For example, maybe you want a wargame that isn't also flagged thematic and that has bag drawing as a mechanism. Or you want a party game that is strategic and uses drafting. Etc etc.
What are family games? In regards to your question, i honestly don't use categories a lot in BGG, but the few times i did, it was useful.
Family games are defined as games that young children and adults can enjoy. For example, Crokinole.
In practice, this category is used for "gateway" games, like Carcassonne.